Hua's lemma

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In mathematics, Hua's lemma,^[1] named for Hua Loo-keng, is an estimate for exponential sums.

It states that if P is an integral-valued polynomial of degree k, $\varepsilon$ is a positive real number, and f a real function defined by

$f(\alpha )=\sum _{{x=1}}^{N}\exp(2\pi iP(x)\alpha ),$

then

$\int _{0}^{1}|f(\alpha )|^{\lambda }d\alpha \ll _{{P,\varepsilon }}N^{{\mu (\lambda )}}$ ,

where $(\lambda ,\mu (\lambda ))$ lies on a polygonal line with vertices

$(2^{\nu },2^{\nu }-\nu +\varepsilon ),\quad \nu =1,\ldots ,k.$

References

↑ Hua, Loo-keng (1938). "On Waring's problem". Quarterly Journal of Mathematics 9 (1): 199–202. doi:10.1093/qmath/os-9.1.199.

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